Shape and strong shape equivalences

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

L UCIANO S TRAMACCIA

Shape and strong shape equivalences

Cahiers de topologie et géométrie différentielle catégoriques, tome 43, n^o4 (2002), p. 242-256

<http://www.numdam.org/item?id=CTGDC_2002__43_4_242_0>

© Andrée C. Ehresmann et les auteurs, 2002, tous droits réservés.

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SHAPE AND STRONG SHAPE

EQUIVALENCES by

^{Luciano STRAMACCIA}

CAHIERS DE TOPOLOGIE ET

GEOMETRIE DIFFERENTIELLE CATEGORIQUES

Volume XLIII-4 (2002)

RESUME. Les concepts

d’6quivalences

^{de forme}

(shape)

^{et de}

forme forte ont leur propre int6r8t

ind6pendamment

de la Th6orie de la Forme elle-même. Ils peuvent 8tre définis dans le cadre abs- trait d’une

paire (C,K)

^de

categories,

^{ou C est}

6quippde

^{d’un fonc-}

teur

cylindre engendrant.

Li6 a leur etude est le

problème

^{de carac-}

t6riser les

6pimorphismes

^et

monomorphismes d’homotopie

^{dans C.}

Pour le rdsoudre, l’auteur utilise la construction de

1’application

cy- lindre double et il introduit une

propri6t6

d’extension

d’homotopie

forte. Il étudie leurs connexions ^avec les concepts

pr6c6dents.

0 INTRODUCTION

’

Shape ^and Strong Shape ^{can be viewed as} theories that allow to approximate general topological spaces by ^{means of} systems ^of spaces that have ni e ho-

motopical properties, ^{such as absolute} neighborhood ^retracts

(ANR-spaces).

The shape category

Sh(Top) [12]

^{has objects all} topological spaces, while a

shape morphisme : ^{X --+ Y can be} represented ^{as a} natural transformation

O- :

[Y, -]

^-

[X, -],

^where

[X, -] :

ANR - Set is the functor which asso-

ciates with _every ANR-space P, ^{the set}

[X, P]

^{of homotopy} classes of maps X - P. A shape equivalence ^{is a} continuous _map inducing ^an isomorphism

in the shape category. ^{Since the} concept ^of strong shape ^{is more} geometrical

in nature, the construction of the strong shape category

SSh(Top)

^{is much}

more involved than the previous ^one and, consequently, ^it gives ^{rise to a}

more interesting concept ^of strong shape equivalence.

In this note we deal with a pair ^of

(C, K)

^{of abstract} categories, ^{where C is}

endowed with a generating cylinder ^functor

[9]

^{and K is the} subcategory ^of

models, ^which plays ^{the role of the} category ^{ANR in the} topological ^case.

In such a context it is possible ^{to define} shape ^and strong shape equivalences internally, ^without going through ^a construction of a shape ^or strong shape category

(see

definitions 1.1 and

1.5).

^{This is done} in section 1, ^{where we}

also introduce a strong version of the homotopy ^extension property

(def-

inition

1.6),

^which turns out to be useful in characterizing strong shape equivalences.

The strong shape category ^of compact metrizable _spaces

SSh(CM)

^{can be}

obtained by localizing ^{CM at the class of} strong shape equivalences, ^while

the localization of CM at ordinary shape equivalences ^{is not} equivalent

to the ordinary shape category

Sh(CM) ([4],

^{see also}

[3]).

^{More recently}

in

[13],

^the strong shape category

SSh(pro Top),

^{for the} category ^{of in-}

verse systems ^of topological spaces, has been defined localizing pro Top

at its class of strong shape equivalences between inverse systems ^of spaces,

as defined in

[11].

Although

SSh(pro Top)

^contains

SSh(Top)

^{as a full} subcategory, ^the arguments ⁱⁿ

[13]

^{do not seem to} give ^a representation ^of

SSh(Top)

^{as the} localization of Top ^at strong shape equivalences. ^{We feel}

that the material contained in this paper could be of help ⁱⁿ studying ^such

kind of problems.

In the second section we consider the double mapping cylinder construction in order to obtain further characterizations of strong shape equivalences, ^also generalizing ^{some results from}

[6]

^and

[7].

The abstract homotopy ^structure given by ^a cylinder, ^or cocylinder, ^{allows one to} study homotopy analogues

of notions such as epimorphisms ^and monomorphisms. For instance

[8]

^in-

troduced the notion of homotopy epimorphism : f : ^{X - Y is a} homotopy epimorphism if, given ho, hl : ^{Y ---+} P, ho f ri hl f implies ^{that ho = hi .}

This corresponds ^to

ip : [Y, P]---+ [X, P]

being ^{1-1. Here for us P will be}

from a subcategory ^{K of the basic} category C, e.g. C ⁼ Top ^{with K =} polyhedra. ^Then this notion of homotopy epimorphism ^will depend ^{on the} subcategory ^K being ^{used. Since a} shape equivalence ^for

(C, K)

^{has to be a}

homotopy epimorphism ^with respect ^{to the} subcategory ^{of models K, then}

we are able ^to shed some light ^{on this} problem, ^also considering ^{the dual}

case of homotopy monomorphism ^and coshape equivalences.

1 PRELIMINARY RESULTS

In what follows C will denote ^a finitely cocomplete category, endowed with

a cylinder ^functor

(I,

^{eo, el ,}

O-).

I : C --+ C is a functor

written I (X)

^{= X x I}

andl(f X - Y) =

^f x1 : X x I ---+ Y x I. _eo,e1

:1C--&#x3E;I ando:I--+1C

are natural transformations such that _O-e0 ⁼ _creel ⁼ id.

We will also assume that the cylinder functor is generating ^{in the sense of}

[7].

Two morphisms f , g : ^{X --+ Y in C are said to be} homotopic, ^written f ri g, if there is a morphism ^{H : X x I -} Y, ^{such that}

Heo

^{= f and}

Hex

^{= g.}

In such a case we say that H is ^a homotopy connecting f and g. The

equivalence ^{class of a} morphism f , generated by ^the homotopy relation, ^will

be denoted by

[f],

^{while the set of} homotopy ^{classes of} morphisms ^from

X to Y will be denoted by

[X, Y].

Consequently, ^{one can define a notion}

of homotopy equivalence in C. In particular, ^{it turns out} that, ^for every X E C, ^the morphisms

ef

^and

e1X

^{are homotopy} equivalences.

Let now fix a full subcategory ^K of C. Then :

- a morphism f : X --+ Y has the homotopy ^extension property

(HEP)

with respect ^to K, ^{whenever the} diagram

is a weak pushout ^{w.r. to K. This means} that, ^for every § : V ---+P ^{P E} K,

and homotopy ^{F : X x} I ---&#x3E; P such that

Fe0x

^{= Of ,} there exists a homotopy

G: Y x I--+ P with GeY0 = ø and G(f ^x1)=F.

A morphism f ^{is called a} cofibration when it has the HEP w.r. to C.

- the mapping cylinder

M( f )

^{of a} morphism f : X --+ Y is given ^by

the following pushout diagram

There is ^an induced morphism fl :

M( f) -

Y, ^{such that}

f1jf =

^{ly and}

f1 II f - fO-x.

^{It is} easily ^shown

[9]

^that f ^{can be} decomposed ^{both as}

f ⁼ f 1 f0 ^and f ⁼

f1 f’0 ,

^{where fi is a} homotopy equivalence ^and f0 ⁼

II fe0X

and

fo

⁼

TIfer

^{are both} cofibrations.

- for every X ^E C, ^let

[X, 2013]: K ---+

^{Set be} the functor that assigns ^to

every P ^E K the set

[X, P].

^Given f : ^{X -} Y, ^{there is an induced natural}

transformation f * :

[Y, -]

^--+

[X, -],

^defined by composition ^with f .

Definition 1.1 A morphism f : ^{X --+ Y in C is said to be a} shape equiv-

alence for the pair

(C, K),

^if f * ^{is a natural} isomorphism.

It follows that f ^{is a} shape equivalence whenever, ^for every P E K, ^the following ^two properties ^{hold :}

(se.1 ) Ip : [Y, P] --+ [X,P]

^{is onto, that is, for} every g : X - P there exists an h : Y - P such that h f ri g.

(se.2) Ip : [Y, P] --+ [X, P]

^{is 1-1, that is,} given ho, hl : ^{Y - P with} ho f ri half, ^{then ho ri hl .}

In the sequel, ^{we shall reserve the name of extensor for a} morphism f : ^{X -}

Y satisfying ^condition

(se.1 ) .

^In

[6]

^such morphisms ^{were called semiequiv-}

alences. Condition

(se.2)

^{states that} f ^{is a} homotopy epimorphism

[8].

We point ^out that both the notions are relative to the pair

(C, K),

although

we shall often omit to indicate it explicitly.

Proposition ^{1.2 The class of extensors for}

(C, K)

^{has the} following prop- erties :

1. Contains all homotopy equivalences.

2. It is closed under composition.

3. If ^a composition g f ^{is an} extensor, ^then f ^{is a extensor.}

4. If f : ^{X - Y is an extensor and}

f’ =

I, ^then

f’ :

X ---+ Y is also an

extensor.

5. f ^{is an extensor iff} f o ^is.

6. f ^{x 1 is an extensor iff} f ^is.

7. If f : X --+ Y is an extensor and has the HEP w.r. to K then, ^for

every g : X --+ P, ^{P E} K, there exists an h : Y -- + P such that h f ⁼ g.

Proof.

(1)

is obvious. Both assertions

(2)

^and

(3)

^follow from the observa- tion that, given f : ^{X - Y} and g : Y --&#x3E; Z, ^then

(g f )*

⁼ f *g* .

(4)

^{For any}

g : X - P, ^{P E} K, ^there is an h : Y --+ P such that h f ri g. Since h f ri h f’,

the assertion follows.

(5).

^Since f ⁼ f1 f0 ^and fi ^{is a} homotopy equivalence,

the assertion follows from the previous ^ones.

(6).

^{Let us observe that, for}

every P E K, ^{there is a} commutative diagram

Since both

(et)p

^and

(eô)p

^are bijections, ^{it follows that}

( f

^x

1) p

^{is onto}

iff f*p)

^{is onto.}

(7).

See, e.g.,

[9],

pag.ll.

Proposition ^{1.3 The class} of homotopy epimorphisms ^for

(C, K)

^{has the} following properties :

1. Contains all homotopy equivalences.

2. It is closed under composition.

3. If a composition g f ^{is a} homotopy epimorphism, then g ^{is a} homotopy epimorphism.

4. If f : ^{X - Y is a} homotopy epimorphism ^and

f’ =

f , ^then

f’ :

^{X ---+ Y}

is also a homotopy epimorphism.

5. f ^{is a} homotopy epimorphism ^iff fo ^is.

6. f ^{x 1 is a} homotopy epimorphism iff f is.

Proof. Let us show part

(5).

^Let f ^{be a} homotopy epimorphisms ^{and let}

vo, VI :

M ( f ) ---+ P,

^{P E} K, ^{be such that} v0 f0 ⁼ v1 f0. Then,

v0 f1 f1 f0 =

V1 f1f1f0,

being f 1

^the homotopy ^{inverse of} f1. Hence, v0 f1 =

vl f 1,

^from

which it follows _{vo ri} vl . Conversely, ^let fo ^{be a} homotopy epimorphisms ^and

let ho, hl : ^{Y -} P, ^{P E} K, ^{be such that} h0 f = half. Then, h0 f1 f0 = hl fl f o implies h0 f1 = h1 f 1, ^hence h0= h1, being f l ^a homotopy equivalence.

From the two propositions ^{above we} obtain the following

Theorem 1.4 The class of shape equivalences ^for

(C, K)

^{has the} following properties:

1. Contains all homotopy equivalences.

2. It is closed under composition.

3. If two of _{the maps} f , g, g f , ^are shape equivalences, ^{so is the third.}

4. If f : ^{X - Y is a} shape equivalence ^and

f’ ci

f , ^then

f’ :

^{X ---+ Y is}

also a shape equivalence.

5. f ^{is a} shape equivalence ^iff fo ^{is a} shape equivalence.

6. f ^{x 1 is a} shape equivalence ^iff f ^{is such.}

Proof. We only ^show

(3).

^Let f : ^{X - Y} and g : ^{Y - Z. It follows}

from the two preceding propositions ^that shape equivalences ^{are closed un-}

der composition. ^Let g f ^and f ^be shape equivalences. ^{It is clear} that g

is then a homotopy epimorphism. ^{Let us show that it is also an extensor.}

Given t : Y - P, ^{there is a v : Z - P such that} vg f = t f . ^Since f ^{is a} homotopy epimorphism, ^{the assertion follows. Let now} g f and g ^be shape equivalences ^{and let} ho, hl : ^{Y -} P, ^{P E K be such that} ho f ri hl f . ^There

are morphisms to, t1, : Z --+ ^{P with} tog f ri ho f ^and ti g f ri hl f . ^{Since then}

to ⁼ tl and g ^{is a} shape equivalence, ^{it follows that} ho ri hl.

We remark that ^a morphism f ^{is a} shape equivalence ^iff fo

(and fo)

^{is also}

a shape equivalence. Hence, ⁱⁿ studying shape equivalences ^we may restrict ourselves to consider morphisms ^{that are} cofibrations. Moreover, ^{in such a} context, ^condition

(se.1 )

^{can be} substituted by ^the stronger ^one expressed by

1.2(7).

The definition of strong shape equivalence ^{is more} involved and we need

some informations in order to give ^it.

For objects X ^{E C and P E} K, ^we shall denote by

7rPx,

the fundamental

groupoid ^{of P under} X, ^as defined, e.g., in

[2].

^{Let us recall that the}

objects ^of -7rPx ^are the morphisms X --+ P in C, ^{while a} morphism ^{cx =}

{G}

^E

nPX(h0,h1),

^{written a : ho} ==&#x3E; hI ^{is a} track from ho ^to hl . ^This

means that cx is the equivalence ^{class of G : X x I -} P, ^with respect ^to the following equivalence relation : G = G’ iff there is ^a homotopy ^{r :}

Y x I x I - P

(rel

^end

maps),

^{that is} having ^the following properties:

Conditions

(3)

^and

(4)

say, respectively, ^that

r(eo

^x

¹⁾

^{is the constant}

homotopy ^at ho ^{and that}

r(ey

^x

1)

^{is the constant homotopy at hl.}

Let us note that, ^for every f : ^{X - Y in} C, ^{there is an induced mor-} phism

(functor)

^of groupoids

f lp

^{: 7rPy --+}

^7rPx

^defined by h t---4 h f ^and

{G} ---+ {G(f

^x

1) }.

Furthermore, homotopic morphisms f=

f’ :

^{X - Y}

induce morphisms ^of groupoids

f# p

^and

f#p ,

^{which are} naturally isomorphic

as functors

([2],

pag.

240).

Definition 1.5 A morphism f : ^{X --+ Y is called a} strong shape equiva-

lence for

(C, K)

^{if it is an extensor} and, moreover,

f# p :

TrPy ---+ 7rPx is full,

for _every P E K.

The condition that

f#p

^{is full, for every P E K, is} equivalent ^to f being

a strong homotopy epimorphism ^{in the} following ^sense: given ^{in C} morphisms ho, h, : ^{Y ---+} P, ^{P E} K, ^{and a} homotopy ^{G : X x I --+} P, ^{G :} hof= hl f , ^{there exists a} homotopy H : ^{Y x I ---+} P, ^{H :} ho ri hl , ^{such that}

H( f

^x

1)

^{and G are} homotopic ^{rel end} maps. In case C is the category Top of topological ^{spaces and} f ^{is a} cofibration then, by

([2], 7. 2. 5 ) ,

^{the above is} equivalent ^to say that

H( f

^x

1)

^{= G.}

Such considerations lead us to give ^the following

Definition 1.6 A morphism f : ^{X - Y in C is said to have the} strong homotopy ^extension property

(SHEP) (w.r.

^to

K),

^{if the} following diagram

is a weak colimit

(w.r.

^to

K)

It follows that in Top ^a cofibration is ^a strong homotopy epimorphism ^{iff it}

has the SHEP. Hence ^a _map which is both a cofibration and an extensor is

a strong shape equivalence ^iff it has the SHEP.

Theorem 1.7 The class of strong shape equivalences ^for

(C, K)

^{has the} following properties:

1. Contains all homotopy equivalences.

2. It is closed under composition.

3. If f : ^{X - Y is a} strong shape equivalence ^and

f’ ri

I, ^then

f’

^{is also}

a strong shape equivalence.

4. f ^{is a} strong shape equivalence ^iff fo ^is.

Proof. It is clear that the composition ^of strong shape equivalences ^{is a} strong shape equivalence. Moreover, ^the Vogt’s ^lemma

[16]

^{implies that}

every homotopy equivalence ^{is a} strong shape equivalence.

(3). f’

^{is an}

extensor by

1.2(4).

^Since

f# p

^{is full, for every P E K, and}

f’p

^{is naturally}

isomorphic ^to

flp,

^it follows that it is full as well.

(4).

f l ^{is a} homotopy equivalence, ^{hence a} strong shape equivalence. ^If fo ^{is also a} strong shape equivalence, ^it follows that f ^is. Conversely, ^let f ^{be a} strong shape equiv-

alence. From f ⁼ fl fo, ^{one has} f0=

fl f,

^where

fl

^{is the homotopy inverse}

of f1. ^{The assertion} then follows from

(3)

^{above and}

1.2(5).

2 THE DOUBLE MAPPING CYLINDER CONSTRUCTION

With the same notations as in the previous section, ^{let us} consider the

following diagram ^{in C}

Its colimit is an object

DM( f , g)

^of C, equipped with three morphisms j0 : ^{Y --+}

DM( f , g),

^{j1 : Z -}

DM( f , g)

^and

.K f,g :

^{X x} I ---&#x3E;

DM( f , g),

such that

KJ,ge6-

^{= jo f ,}

K f,ge1X

^{= jig, and having} the suitable universal property

[9].

^{It is called the double} mapping cylinder ^{of the} pair

( f , g ) .

In case the two morphisms f and g coincide, ^{then we} speak ^{of the double} mapping cylinder ^of f ^{and write} simply

DM( f )

^for

DM( f , f )

^and

K f

^for

Kf,f.

It is worth to point ^{out that the} triple

(DM( f , g), j0, j1 )

^{is actually the} homotopy pushout ^{of the} diagram

Y +---f

X ---+ g Z and

DM( f )

^{is the case}

where the two morphisms ^{involved are the same. The situation can be better}

illustrated by ^the following diagram

where the two squares ^are commutative and, ^for every triple

(ho,

^hl;

H),

h0, h1 : Y ---+ P, ^{P E} C, and H : X x I --+ P a homotopy connecting ho f ^with hl f , there exists a unique morphism, :

DM( f ) ---+

P, ^{such that} -yK = ^{H and} yjo ⁼ h0 ’Yjl ⁼ he.

Let ^us note that there is a unique morphism (D :

DM(f) ---+YxI

^{such that}

4PKf

⁼ f ^{x 1} and CP jo ⁼

e0y, CP j1

⁼

eiY .

Remark 2.1 If f : X - Y is a continuous _map, then

DM( f )

^{may be}

described as the ad,junction space

(X

^x

I )

U f

(Y

^x

81),

^{where 81 =}

{0, 1}.

^If f ^{is an inclusion} map, then

D M ( f )

^{is the} subspace

(X x I) U (Y x aI)

^{of Y x I.}

In case f ^{is a}

(closed)

cofibration, ^the inclusion lF: X x I U Y x aI ---+ Y x I is also ^a closed cofibration

([14], Th.6).

A discussion of the double mapping cylinder ^{of a} continuous _map ^can be found in

[5], [15]

^{and, more recently, in}

[9],

^{see also}

[6], [10]

^and

[11].

A version of the following ^{theorem was} proved ⁱⁿ

[6],

^{Thm. 5.2, for C the}

category ^of compact ^metric spaces and proper maps and K ⁼ ANR.

Theorem 2.2 If 4l :

DM(f) --+ Y x I is an

^{extensor, then f is a strong} homotopy epimorphism.

Proof. Let ho, hl : Y - P, P E K, and let H : X ^x I - P be a homotopy connecting ho f ^and hl f . ^{From the double} mapping cylinder diagram ^one

obtains a morphism, :

DM( f ) ---+

^{P such that} qjo ⁼ ho, yj1* ⁼ hl ^and

-yKf

^{= H. Since F =}

pF(HF e0DM(f))

^{is an extensor, then}

HF eoDM(f )

^{is itself}

an extensor, by Prop.

1.2(3).

^{Since it is also a} cofibration, ^from Prop.

1.2(7),

it follows that there exists a G :

M(lF) --+

^{P such that}

GII lFe0 DM (f) =y.

^From

the commutative diagram

it follows that G jlF ^{is a} homotopy connecting ho ^and hl . ^{In fact :}

Hence, f ^{is a} homotopy epimorphism. Finally, ^{let us} observe that

G jlF( f

^x

1)

⁼

Gj lF lF K f

⁼

GII lF eo DM(f ) Kf

⁼

-yK f

^{= H, from which it follows that}

0

is full, ^for every P E K.

Corollary ^{2.3 If} f : ^{X - Y and lF :}

DM( f) -

^{Y x I are} extensors, ^then

f ^{is a} strong shape equivalence.

Proposition ^{2.4 Let} f : ^{X ---+ Y be a} morphism ^{in C. If} f ^{has the SHEP}

w.r. to K, then lF is an extensor for

(C, K).

Proof. Let _{p :}

DM( f) -

P, ^{P E K.} Then, ^{there is a} morphism ^{V :}

Y x I - P such that

V (f

^x

1)

⁼

pKf

^and

Vey =

pji, ^{i =} 0,1. ^Since

V4)Kf

⁼

V(f

^x

1)

⁼

pKf

^{and V(Dji =}

YeY

^{= pji , i = 0,1, by} the universal

property of the double mapping cylinder, ^it follows that VlF ⁼ _p.

It is clear that an extensor f : ^{X ---+ Y is a} strong shape equivalence ^whenever

either lF is an extensor or f has the SHEP ^w.r. to K. These last two facts turn out to be equivalent ^{in the} category ^of topological spaces, in fact, together with the above proposition, ^{there the} following theorem holds : Theorem 2.5 Let f : ^{X - Y be a} cofibration in Top. ^{If lF :}

DM( f ) ---+

Y x I is an extensor for

(Top, K),

^then f has the SHEP w.r. K.

Proof. Given _maps ho, ^{h, :} Y ---&#x3E; P, ^{P E} K, ^{and a} homotopy ^{H :} X x I ---+ P, connecting ho f and half, there exists a map, :

DM(f) ---+

^{P, such that}

-yKf = H and

^{qjx = ha, A = 0,1. Since (D has the} HEP, by Prop.

1.2(5),

there is a homotopy ^{T, : Y x I ---+ P such} that TlF = _q.

We obtain the following relations : i.

r( f

^x

1)

⁼

TlF Kf

⁼

qKj

^{= H,}

2.

rer =

^{TlF jx = yjx = ha, A = 0, 1,}

from which it follows that F is a homotopy connecting ho ^{and hl.}

Corollary ^{2.6 Let} f : ^{X--+ Y be a} cofibration in Top. ^The following properties ^are equivalent:

1. f ^{is a} strong homotopy epimorphism.

2. f ^{has the SHEP.}

3. lF is an extensor.

Remark 2.7 In view of

1.4(4)

^and

1.7(3),

^the assumption ^that f ^{has to be}

a cofibration is almost harmless.

Proposition ^{2.8 Let} f : ^{X - Y be an extensor such that} f ^{x 1 is a}

cofibration. Then f ^{is a} strong shape equivalence ^{iff it has the SHEP w.r.}

to K.

Proof. Let ho, hl : Y ---+ P, ^{P E} K, ^{and let H : X x I --+ P be a} homotopy connecting ho f ^with h1f . ^If f ^{is a} strong shape equivalence, ^{there is a homo-} topy G’ : h0= hl ^{such that}

G’( f

^x

1)=

^{H. Since} f ^{x 1 is an extensor and a} cofibration, ^then there exists a homotopy ^{G = G’ such that}

G( f

^x

1) -

^H.

Finally, ^since f ^{is a} homotopy epimorphism, ^{it follows that G :} h0= hl.

Example ^{2.1 Let Met be the} category of metrizable _spaces and Ar be the full subcategory of absolute retracts for metrizable _spaces. Every ^closed embedding f : ^{X - Y in Met is a} strong shape equivalence ^{w.r. to Ar. In} fact, every closed embedding of metrizable _spaces is an extensor

([1],

^pag.

87)

^and has the HEP w.r.to Ar

(this

^is the Borsuk’s homotopy ^extension theorem,

[1],

pag.

94).

^{Note that also $ :}

DM( f )

^{- Y x I is a closed} embedding of metrizable _spaces.

3 THE DUAL CASE

The results of the previous sections, concerning homotopy ^and strong ^ho- motopy epimorphisms, ^can be dualized in order to treat the problem ^of homotopy monomorphisms. ^{We state here the} appropriate ^dual concepts.

Starting again ^{from a} pair ^of categories

(C, K),

^{let us} consider, ^for every X E C, ^the set-functor

[-,X] :

^{K -} Set, ^{P -}

[P, X].

Every morphism f : X-- -&#x3E; Y induces a natural transformation f * :

[-X]--+ [-, Y] .

^The

morphism f ^{will be} called here a coshape equivalence ^whenever f * ^turns

out to be a natural isomorphism. Then, f ^{will be a} coshape equivalence ^if

(cse.l )

^{f is a} retractor, ^that is, ^for every g : P - Y, ^{P E} K, ^{there is}

some h : P ---+ .X such that f h - g.

(cse.2)

f ^{is a} homotopy monomorphism, ^that is, ^for every ho, hl :

P ---+ X, ^{the fact that} f ho - f hl implies ^that ho ri hl .

Note that also these notions are to be considered with respect ^to

(C, K).

We will assume that C is now finitely complete ^and that it is endowed with

a generating cocylinder ^functor

« - )1, fO, fl, B),

^{as defined in}

[9].

^Here

(-)I :

C --&#x3E; C, ^{X -}

X I ,

f -

,f I ,

^and eo, ei :

(-)I

^-

1C, s : 1C ---+ (-)I ,

are natural transformations such thateos = Els ⁼ id. The homotopy ^relation

used here is that induced by ^the cocylinder ^functor.

Let us recall that a morphism f : ^{X ---+ Y is said to have the} homotopy lifting property

(HLP)

^{w.r. to K, whenever the} following diagram ^{is a weak} pullback ^{w.r. to} K,

The mapping cocylinder ^{of a} morphism f : X --+ Y is the object

E( f )

which _appears in the following pullback diagram

Every morphism f : ^{X--+ Y in C has a} mapping cocylinder decomposition f =

fo fl ,

^where

f 1

^{is a} fibration, ^that is, ^it has has the HLP w.r. to C,

and

f0

^{is a} homotopy equivalence.

We have ^now the notion of double mapping cocylinder ^of f , ^{which is}

defined by ^{the limit}

in C of the diagram

Here again ^one should be careful that, ⁱⁿ general, ^one defines the double

mapping cocylinder

DE( f , g)

^{of a pair}

X f---+

^y+--- Z and that

DE ( f )

^is

just ^an abbreviation for the ^case where f and g ^{coincide. Then}

DE(f)

^is

the homotopy pullback ^of

X ---+ Y+---

^X

_[9].

All the results of the previous ^{section can be} dualized for retractors, ^homo- topy monomorphisms ^and coshape equivalences. ^For instance, ^{let us note}

that there is a unique morphism : XI --&#x3E;

DE f ),

having ^the property ^that

vfw

⁼

f I

^{and u0w =}

E0Y ,

^{UIW =}

fro

^{It is} interesting ^to give ^{the dual form}

of Theorem 2.2 and Cor. 2.6, ^{as follows:}

Theorem 3.1 If w : X I ---+

DE(f))

^{is a} retractor, ^then f ^{is a} homotopy monomorphism.

Proposition ^{3.2 Let} f : X ---+ Y be a fibration in Top. ^The following ^are equivalent :

1. f ^{is a} strong homotopy monomorphism.

2. f ^{has the SHLP.}

3. Bl1 is a retractor.

The definition of strong homotopy monomorphism ^and strong homotopy lifting property

(SHLP)

^are the obvious ones.

References

[1]

^K. Borsuk, Theory of Retracts, ^Polish Scientific Publishers, ^Warsawa

1967.

[2]

^R. Brown, Topology, ^Ellis Horwood, ^1988.

[3]

^F.W. Cathey, Strong Shape Theory, ^{Lect. Notes in Math.} 870, Springer Verlag,

(1981),

^215-238.

[4]

^{A. Calder - H.M. Hastings,} Realizing Strong Shape Equivalences, ^J.

Pure Appl. Alg. ²⁰

(1981),

^129-156.

[5]

^{A. Dold, Metodi Moderni di Topologia} Algebrica, Quaderni ^del C.N.R.,

Roma 1973.

[6]

^J. Dydak - ^J. Segal, Strong Shape Theory, Dissertationes Math. CXCII,

Warsawa 1981.

[7]

^{J. Dydak - S.} Nowak, Strong Shape for Topological Spaces, ^{Trans. Am.}

Math. Soc. 323,

(2) (1991),

^765-796.

[8]

^{P.J. Hilton,} Homotopy Theory ^and Duality, Gordon Breach 1965.

[9]

^K.H. Kamps - ^T. Porter, ^Abstract Homotopy ^and Simple Homotopy Theory, ^World Scientific, ^1997.

[10]

^G. Kozlowsky, Images of ANR’s, Mimeographed ^{notes, Univ. of Wash-} ington, Seattle, 1974.

[11]

^S. Mardesic, Strong Shape ^and Homology, Springer Verlag, ^2000.

[12]

^S. Marde0161i0107 - J. Segal, Shape Theory, ^North Holland, ^1982.

[13]

^A. Prasolov, Extraordinary Homology Theory , Top. Appl. ¹¹³

(2000),

. 249-291.

[14]

^{A. Strøm, Note on} Cofibrations II, Math. Scand. 22

(1968),

^130-142.

[15]

^{T. tom} Dieck - K.H. Kamps - ^D. Puppe, Homotopietheorie, ^{Lect. Notes}

in Math. 157, Springer Verlag, ^1970.

[16]

^R. Vogt, ^{A Note on} Homotopy Equivalences, ^{Proc. Am. Math. Soc. 32}

(1972),

^627-629.

Dipartimento ^di Matematica e Informatica Universita di Perugia - ^Italia

stra@dipmat.unipg.it